Finite Group Databasebeta

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C₂×D₇×C₃₁₁ #22576534

order	8708
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃₁₁ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×D₇×C₃₁₁
unambiguous semidirect product	C₇⋊(C₂²×C₃₁₁)
unambiguous semidirect product	(C₂×C₇)⋊(C₂×C₃₁₁)

Splittings

Fitting, socle	C₂×C₇×C₃₁₁ ⋊ C₂
derived, Sylow	C₇ ⋊ C₂²×C₃₁₁
Hall	C₇×C₃₁₁ ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃₁₁	D₇
commutator, hypocenter	C₇	C₂²×C₃₁₁
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₃₁₁	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₃₁₁

20 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

3110 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×AGL(1,𝔽₇)×C₃₁₁, D₃×D₇×C₃₁₁, C₂×C₃×D₇×C₃₁₁, …
maximal normal subgroup of	C₂×AGL(1,𝔽₇)×C₃₁₁, C₂×C₃×D₇×C₃₁₁
maximal quotient of	D₃×D₇×C₃₁₁, C₂×C₃×D₇×C₃₁₁, C₂×D₂₁×C₃₁₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(8708, 8);